3.182 \(\int \frac{A+B x}{x^4 (a+b x)} \, dx\)

Optimal. Leaf size=86 \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{3 a x^3} \]

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(2*a^2*x^2) - (b*(A*b - a*B))/(a^3*x) - (b^2*(A*b - a*B)*Log[x])/a^4 + (b^2*(A*b -
a*B)*Log[a + b*x])/a^4

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Rubi [A]  time = 0.0506325, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {77} \[ -\frac{b^2 \log (x) (A b-a B)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{A}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/(x^4*(a + b*x)),x]

[Out]

-A/(3*a*x^3) + (A*b - a*B)/(2*a^2*x^2) - (b*(A*b - a*B))/(a^3*x) - (b^2*(A*b - a*B)*Log[x])/a^4 + (b^2*(A*b -
a*B)*Log[a + b*x])/a^4

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int \frac{A+B x}{x^4 (a+b x)} \, dx &=\int \left (\frac{A}{a x^4}+\frac{-A b+a B}{a^2 x^3}-\frac{b (-A b+a B)}{a^3 x^2}+\frac{b^2 (-A b+a B)}{a^4 x}-\frac{b^3 (-A b+a B)}{a^4 (a+b x)}\right ) \, dx\\ &=-\frac{A}{3 a x^3}+\frac{A b-a B}{2 a^2 x^2}-\frac{b (A b-a B)}{a^3 x}-\frac{b^2 (A b-a B) \log (x)}{a^4}+\frac{b^2 (A b-a B) \log (a+b x)}{a^4}\\ \end{align*}

Mathematica [A]  time = 0.0499314, size = 81, normalized size = 0.94 \[ \frac{\frac{a \left (a^2 (-(2 A+3 B x))+3 a b x (A+2 B x)-6 A b^2 x^2\right )}{x^3}+6 b^2 \log (x) (a B-A b)+6 b^2 (A b-a B) \log (a+b x)}{6 a^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/(x^4*(a + b*x)),x]

[Out]

((a*(-6*A*b^2*x^2 + 3*a*b*x*(A + 2*B*x) - a^2*(2*A + 3*B*x)))/x^3 + 6*b^2*(-(A*b) + a*B)*Log[x] + 6*b^2*(A*b -
a*B)*Log[a + b*x])/(6*a^4)

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Maple [A]  time = 0.008, size = 101, normalized size = 1.2 \begin{align*} -{\frac{A}{3\,a{x}^{3}}}+{\frac{Ab}{2\,{a}^{2}{x}^{2}}}-{\frac{B}{2\,a{x}^{2}}}-{\frac{{b}^{2}A}{{a}^{3}x}}+{\frac{bB}{{a}^{2}x}}-{\frac{A\ln \left ( x \right ){b}^{3}}{{a}^{4}}}+{\frac{{b}^{2}B\ln \left ( x \right ) }{{a}^{3}}}+{\frac{{b}^{3}\ln \left ( bx+a \right ) A}{{a}^{4}}}-{\frac{{b}^{2}\ln \left ( bx+a \right ) B}{{a}^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/x^4/(b*x+a),x)

[Out]

-1/3*A/a/x^3+1/2/a^2/x^2*A*b-1/2/a/x^2*B-1/a^3*b^2/x*A+1/a^2*b/x*B-1/a^4*b^3*ln(x)*A+1/a^3*b^2*ln(x)*B+1/a^4*b
^3*ln(b*x+a)*A-1/a^3*b^2*ln(b*x+a)*B

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Maxima [A]  time = 1.03134, size = 120, normalized size = 1.4 \begin{align*} -\frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (b x + a\right )}{a^{4}} + \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left (x\right )}{a^{4}} - \frac{2 \, A a^{2} - 6 \,{\left (B a b - A b^{2}\right )} x^{2} + 3 \,{\left (B a^{2} - A a b\right )} x}{6 \, a^{3} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="maxima")

[Out]

-(B*a*b^2 - A*b^3)*log(b*x + a)/a^4 + (B*a*b^2 - A*b^3)*log(x)/a^4 - 1/6*(2*A*a^2 - 6*(B*a*b - A*b^2)*x^2 + 3*
(B*a^2 - A*a*b)*x)/(a^3*x^3)

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Fricas [A]  time = 1.94261, size = 203, normalized size = 2.36 \begin{align*} -\frac{6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (b x + a\right ) - 6 \,{\left (B a b^{2} - A b^{3}\right )} x^{3} \log \left (x\right ) + 2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="fricas")

[Out]

-1/6*(6*(B*a*b^2 - A*b^3)*x^3*log(b*x + a) - 6*(B*a*b^2 - A*b^3)*x^3*log(x) + 2*A*a^3 - 6*(B*a^2*b - A*a*b^2)*
x^2 + 3*(B*a^3 - A*a^2*b)*x)/(a^4*x^3)

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Sympy [B]  time = 0.93392, size = 165, normalized size = 1.92 \begin{align*} \frac{- 2 A a^{2} + x^{2} \left (- 6 A b^{2} + 6 B a b\right ) + x \left (3 A a b - 3 B a^{2}\right )}{6 a^{3} x^{3}} + \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} - a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} - \frac{b^{2} \left (- A b + B a\right ) \log{\left (x + \frac{- A a b^{3} + B a^{2} b^{2} + a b^{2} \left (- A b + B a\right )}{- 2 A b^{4} + 2 B a b^{3}} \right )}}{a^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x**4/(b*x+a),x)

[Out]

(-2*A*a**2 + x**2*(-6*A*b**2 + 6*B*a*b) + x*(3*A*a*b - 3*B*a**2))/(6*a**3*x**3) + b**2*(-A*b + B*a)*log(x + (-
A*a*b**3 + B*a**2*b**2 - a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4 - b**2*(-A*b + B*a)*log(x + (-A*a
*b**3 + B*a**2*b**2 + a*b**2*(-A*b + B*a))/(-2*A*b**4 + 2*B*a*b**3))/a**4

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Giac [A]  time = 1.31399, size = 134, normalized size = 1.56 \begin{align*} \frac{{\left (B a b^{2} - A b^{3}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} - \frac{{\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b x + a \right |}\right )}{a^{4} b} - \frac{2 \, A a^{3} - 6 \,{\left (B a^{2} b - A a b^{2}\right )} x^{2} + 3 \,{\left (B a^{3} - A a^{2} b\right )} x}{6 \, a^{4} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/x^4/(b*x+a),x, algorithm="giac")

[Out]

(B*a*b^2 - A*b^3)*log(abs(x))/a^4 - (B*a*b^3 - A*b^4)*log(abs(b*x + a))/(a^4*b) - 1/6*(2*A*a^3 - 6*(B*a^2*b -
A*a*b^2)*x^2 + 3*(B*a^3 - A*a^2*b)*x)/(a^4*x^3)